Applied bias at Zero Kelvin?

Hypothetically:
A pair of electrodes with a small insulating barrier between them. Transport is via tunneling through the barrier.
You apply a small bias, and measure conductance because there is tunneling current.
At 0 K, can there still be tunneling?
Is it possible for a tunnel junction to be AT 0 K while there is an applied bias, or does the very nature of the 'potential' mean that it is at some T≠0 state?

Sure, you can have tunnelling. The latter is after all a process that does not depend on the availability of thermal energy,

Experimentally, these measurements are often done at temperatures of about 20 mK or so, meaning kbT is nearly always much, much lower than the potential energies involved (if this is not the case, thermal processes will make it difficult to see the tunnelling current).